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In this tutorial, a derivation of magnetohydrodynamics (MHD) valid beyond the usual ideal gas approximation is presented. Non-equilibrium thermodynamics is used to obtain conservation equations and linear constitutive relations. When coupled with Maxwell's equations, this provides closed fluid equations in terms of material properties of the plasma, described by the equation of state and transport coefficients. These properties are connected to microscopic dynamics using the Irving–Kirkwood procedure and Green–Kubo relations. Symmetry arguments and the Onsager–Casimir relations allow one to vastly simplify the number of independent coefficients. Importantly, expressions for current density, heat flux, and stress (conventionally Ohm's law, Fourier's law, and Newton's law) take different forms in systems with a non-ideal equation of state. The traditional form of the MHD equations, which is usually obtained from a Chapman–Enskog solution of the Boltzmann equation, corresponds to the ideal gas limit of the general equations. 

Non-neutral plasma experiments are excellent benchmarks for validating transport models, including in strongly coupled conditions. Experiments with Penning–Malmberg traps operate under the Brillouin limit, which means that the plasma is also strongly magnetized in the sense that the gyrofrequency exceeds the plasma frequency. This is an unusual regime that is not described by traditional plasma kinetic theory, particularly when strong coupling and strong magnetization are both present. Here, we apply a recently developed generalized Boltzmann kinetic theory to compute the temperature anisotropy relaxation rate in this regime. Strong magnetization is found to severely suppress energy exchange during collisions, leading to a drastically reduced anisotropy relaxation rate. The results exhibit good agreement with previous work by Glinsky et al. when the plasma is weakly coupled and extend the calculation to the strongly coupled regime as well. Results are compared with published experimental measurements, demonstrating good agreement. Furthermore, the model is tested using molecular dynamics simulations over a broader range of parameters than the experiments reached. These simulations utilize a new Green–Kubo relation, enabling an equilibrium simulation method that is more accurate than previous non-equilibrium methods that have been applied to this problem. Finally, a discussion of detailed balance in strongly magnetized plasmas is provided. Specifically, it is shown that despite the absence of time-reversal symmetry, which is usually used to mathematically prove detailed balance, the results satisfy detailed balance to a high degree of numerical precision.